小Excess与开流形的拓扑

本文中，我们应用比较几何的方法研究开流形的Excess与其拓扑之间的关系，我们证明了对于一个曲率下有界的开流形，当它的Excess被临界半径的某个函数所界定时，它就有有限拓扑型或微分同胚于n维z欧氏空间。     

径向截面曲率有下界黎曼流形的微分同胚定理

研究了径向截面曲率以一类旋转模曲面的Gauss曲率为下界的非紧完备黎曼流形的拓扑,得到了该类黎曼流形与欧氏空间微分同胚的一个合理的充分条件,推广了径向截面曲率有常数下界完备黎曼流形的微分同胚定理.     

Ricci曲率平行的一类Riemann流形的C∞紧性

本文证明,在Gromov-Hausdorff拓扑下,Ricci曲率平行,截面曲率和单一半径有下界,体积有上界的Riemann流形的集合是c^∞紧的.作为应用,我们证明一个pinching结果,即在某些条件下,Ricci平坦的流形必定平坦.     

具非负Ricci曲率和次大体积增长的流形

设M是具非负Ricci曲率的n维黎曼流形,其截曲率有下界,对M中的任意的点p有vol[B(p,r)]/rn-1=αM+o(1/rn-1)且假设函数f(r)=vol[B(p,r)]/2In(r)rn-1是单调递减的,则M具有限拓扑型,其中In(r)是一有界函数.     

关于H. Wu问题

著名几何学家Ｈ．Ｗｕ在「４」中提出这样的问题：若一完备非紧的黎曼流形仅有两个符号相反的Ｂｕｓｅｍａｎｎ函数,则该流形的结构如何？本文在流形具非负Ｒｉｃｃｉ曲率或截曲率具下界的情况下部分地解决了这一问题。同时还讨论了流形在具非负曲率条件下有关的一些性质。     

Ricci曲率平行的一类Riemann流形的C~∞紧性(英文)

本文证明,在Ｇｒｏｍｏｖ－Ｈａｕｓｄｏｒｆｆ拓扑下,Ｒｉｃｃｉ曲率平行,截面曲率和单一半径有下界,体积有上界的Ｒｉｅｍａｎｎ流形的集合是ｃ∞紧的．作为应用,我们证明一个ｐｉｎｃｈｉｎｇ结果,即在某些条件下,Ｒｕｃｃｉ平坦的流形必定平坦．     

Finsler流形上Laplace比较定理及其应用

本文研究了Finsler流形上距离函数的Laplacian.利用Schwarz不等式和[5]中主要方法,获得了具有负曲率的Laplacian比较定理,进而得到了Finsler流形上第一特征值的下界估计.     

Finsler流形上的Laplace比较定理和有界估计(英文)

本文研究了Finsler流形上的距离函数的Laplacian.利用指标引理和文献[4]中主要方法,获得了Ricci曲率有函数下界的Laplacian比较定理,改进了文献[6]和文献[7]的相关结果.     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

具有渐近非负Ricci曲率和无穷远处二次曲率衰减的流形

本文首先给出了具有渐近非负Ricci曲率流形的体积比较定理.然后给出了流形在一定的曲率衰减的条件下为有限拓扑型的引理,最后利用Abresch-Gromoll估计,给出了具有渐近非负Ricci曲率和无穷远处二次曲率衰减的流形的有限拓扑型条件.     

小Excess与具负下界Ricci曲率开流形的拓扑

In this paper,we study the relation between the excess of open manifolds and their topology by using the methods of comparison geometry.We prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its Excess is bounded by some function of its conjugate radius,which improves some results in [4].     

Hyperbolization of cusps with convex boundary

We prove that for every metric on the torus with curvature bounded from below by ?1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.     

Open manifolds with nonnegative Ricci curvature and large volume growth

In this paper, we study complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We prove among other things that such a manifold is diffeomorphic to a Euclidean n-space if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in . Received: August 17, 1998.     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.     

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Science China Mathematics - Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is...     

Complete gradient shrinking Ricci solitons have finite topological type

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:(i) the Ricci curvature is bounded from above;(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. To cite this article: F.-q. Fang et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Ricci Flow of Regions with Curvature Bounded Below in Dimension Three

We consider smooth complete solutions to Ricci flow with bounded curvature on manifolds without boundary in dimension three. Assuming an open ball at time zero of radius one has sectional curvature bounded from below by ?1, then we prove estimates which show that compactly contained subregions of this ball will be smoothed out by the Ricci flow for a short but well-defined time interval. The estimates we obtain depend only on the initial volume of the ball and the distance from the compact region to the boundary of the initial ball. Versions of these estimates for balls of radius r follow using scaling arguments.     

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

Ricci curvature,radial curvature and large volume growth

In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound, we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover, by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with large volume growth.